Nonlinear PDE aspects of Ricci Flow

 While in discussion with a colleague, I remembered Terry Tao's excellent paper on the nonlinear PDE aspects of Ricci flow and the Poincare conjecture. See Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective. See also his slides on the Poincare conjecture proof: Perelman's proof of the Poincaré conjecture. I personally am fascinated by the Sacks-Uhlenbeck theorem, which has to do with a nontrivial $\pi _{2}(M)$ and shows the finite-time existence of singularities. This finds its way in turn into what Hamilton first showed as a kind of finite-time existence of singularities. It is also interesting how the homotopy cobordism theorem aspects also have implications in a fundamental sense.